lab 2. single photon interference
TRANSCRIPT
Lab 2. Single Photon Interference
Mehul Malik and Pengke Li
November 15, 2007
I. Abstract
Interference experiments are carried out at the single photon level in a Young’s double slit setup and a Mach-
Zehnder interferometer. The dual wave-particle nature of photons is observed. The dependence of visibility
of interference fringes on ”which-path” information is also studied in the Mach-Zehnder interferometer.
II. Introduction
The principle of wave-particle duality stipulates that all matter has both wave and particle properties. The
famous de Broglie hypothesis of 1924 was the first scientific generalization of this principle that related the
momentum of a particle to its wavelength [1]. De Broglie’s principle states that all matter, be it a tennis
ball or an atom, has a wave-like nature, with a wavelength described by the simple formula,
λ =h
p(1)
where p is the particle’s momentum and h is planck’s constant. The dual wave and particle nature of photons
had been confirmed much before de Broglie’s hypothesis with Young’s double slit experiment that showed
interference of two beams of light, and with Einstein’s hypothesis of the photoelectric effect and subsequent
experiments. The question of duality was tackled by the Copenhagen interpretation of quantum mechanics,
within which Bohr and Heisenberg proposed their famous complementarity principle. The complementarity
principle stated that while matter had both wave and particle properties, an experiment could not measure
both properties at the same time.
A problem arose when one tried to explain the wave nature of light at the single particle level. If a
source of particles were to be attenuated so that only one particle at a time were to reach the slits in a
Young’s experiment, then how could one conceive of the single particle giving rise to interference effects?
The answer to this question was laid out quite clearly by Dirac. Within the Copenhagen interpretation, it
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can be said that before photons are measured, they are described by their wave functions. Dirac postulated
that wave function gives the probability of each photon being in a particular place and not the probable
number of photons in that place, as was believed earlier [2]. Thus, when the wave function that describes
a photon ”approaches” two slits that are within a wavelength of each other, the probabilistic photon goes
through both slits and essentially interferes with itself. While this is a strange concept to visualize, it is
important to understand that within the Copenhagen interpretation of Quantum Mechanics, there is no
deterministic reality before measurement, one can only describe the photon in terms of the wave function,
which is probabilistic in nature.
Another way to understand the idea of a photon interfering with itself is in the Mach-Zehnder inter-
ferometer, which involves sending a photon through a beam splitter down two paths, and then varying one
of the path lengths. According to Dirac’s interpretation [2], the photon probabilistically travels in both
paths and hence can be said to be interfering with itself. Another way to explain this is that there is no
”which-path” information available. If we try to measure which path the photon took, the probabilistic wave
function that is present in both paths collapses to one of the paths and the interference effects disappear. It
may be said then that the photon exists only after it is measured - though that idea is the subject of much
philosophical debate.
III. Experimental Setup and Procedure
In our experiment, the single photons are generated by attenuating a CW HeNe laser at 632.9 µm to the
extent that there is only one photon in the path at a time. This is done by measuring the laser power and
dividing by the energy of a photon, which gives the number of photons per second coming out of the laser.
Then, the time spent in the experimental setup is calculated by dividing the total path length by the speed
of light. That time is then multiplied by the rate of photons to give the total number of photons in the
experimental path at a time. The total attenuation required is simply the inverse of this number, which is
experimentally obtained using neutral density filters of varying magnitudes. In the case of Young’s double
slit experiment, the attenuation required is calculated to be 1.96×10−5, which is obtained by grouping filter
numbers 1, 1 OMA and 2 OMA together. In the case of the Mach-Zehnder interferometer, the attenuation
required is calculated to be 5.35×10−6, which is obtained by grouping filter numbers 5, 1 OMA and 3 OMA
together. In this manner, one photon at a time is obtained in the experimental path. A cooled EM-CCD
camera is used to observe interference fringes.
In the case of the double slit experiment shown in figure 1, the aquisition time of the camera will be
gradually decreased which corresponds to fewer and fewer single photons making it through the slits. This
should result in a reduced visibility of interference fringes. It should also be possible to observe single events
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He-Ne Laser
Neutral Density Filter(s)
Mirror
Spatial Filter
CCD Camera
Screen with two slits (greatly exaggerated for clarity)
Cone of light due to diffraction
Image at CCD array
Figure 1: Young’s double slit experiment setup
that indicate discrete particle-like behavior of the photons by optimizing the gain and aquisition time. In
the Mach-Zehnder interferometer, the light incident on the interferometer is linearly polarized at 45◦. As
shown in figure 2, the polarizing beam-splitter A then directs the 0◦ (H) component of the 45◦ photon
into one arm and the 90◦ (V) component into the other arm. These orthogonal components meet at the
second beam-splitter. A second polarizer D projects the output photon onto a new basis determined by the
polarizer angle. If the resultant photon is then measured in the H or V basis, the ”which-path” information
of the photon is preserved and no interference is observed. If the measurement is carried out in the 45◦ or
(H − V )/√
2 basis, then the ”which-path” information is erased. It can be said that the polarizer allows
the 45◦ components of both the H and the V polarization states through, thus allowing interference to be
observed.
He-Ne Laser
Neutral Density Filter(s)
Mirror
Mirror
Spatial Filter
Polarizer A Polarizing Beam Splitter
Non-polarizing Beam Splitter
Polarizer D
Screen
CCD Camera
Path 1 (Horizontal Polarization)
Path 2 (Vertical Polarization)
Figure 2: Mach-Zehnder interferometer setup
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If we were to explain this in terms of the Copenhagen interepretation, we would say that the wave
function of the photon is actually in a superposition of all polarization states. A polarizer simply modifies the
probability distribution to have a higher amplitude at the set polarizer angle. Hence for crossed polarizers, we
get a sin2θ dependence over time on relative polarizer angle θ. The H and the V states in the Mach-Zehnder
interferometer can be said to be in equal superpositions of 45◦ and −45◦ states:
|H〉 =1√2
(|+ 45◦〉+ | − 45◦〉)
|V 〉 =1√2
(|+ 45◦〉 − | − 45◦〉) (2)
The polarizer at 45◦ modifies both wave-functions to give a higher probability of 45◦ states, and hence
intereference fringes are observed. As the polarizer is rotated to the H polarization basis, it modifies the
wave-function of any incident photon to give a maximum probability amplitude for the H component and a
minimum for the V component, thus making it impossible for the photon to probabilistically interfere with
itself.
IV. Experimental Results
A. Young’s Double Slit Experiment
As is expected, the interference from the double slit setup reduces in visibility as the acquisition time
is decreased and the gain is increased. For an optimal acquisition time and gain, one can clearly see the
Figure 3: Particle-like behavior of single photons (image acquisition time 0.03 sec, gain=100)
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(a) acq. time 1 sec, gain=0 (b) acq. time 0.015 sec, gain=135
(c) acq. time 0.0013 sec, gain=190 (d) acq. time 0.0003 sec, gain=200
Figure 4: Interference in Young’s double slit experiment with decreasing acquisition time
particle nature of the photons as they make themselves manifest as discrete single events in figure 3. As is
seen in figure 4, the interference fringes are washed out as the acquisition time is decreased from 1 second
(with no gain) to 0.0003 seconds (with gain equal to 200).
B. Mach-Zehnder Interferometer
A similar experiment to observe particle-like behavior of the photons is carried out with the Mach-Zehnder
interferometer. The acquisition time is slowly decreased and the fringes are seen to wash out. However,
the more interesting experiment in this case is when the D polarizer is rotated from 45◦ to 90◦. The fringe
visibility is reduced, as is seen clearly in the CCD images in figure 5.
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(a) Polarizer D at 45◦ (b) Polarizer D at 65◦
(c) Polarizer D at 80◦ (d) Polarizer D at 90◦
Figure 5: Fringes in the Mach-Zehnder interferometer for different Polarizer D angles
The fringe visibility for different polarizer D angles as seen along the vertical cross-section of the fringe
images is also plotted in figure 6. The visibility reduction can be seen more quantitatively. This is not
very precise as the fringes are not very uniform horizontally. However, on average, we can see that the
visibility decreases. This confirms that the rotation of polarizer D erases the ”which-path” information in
the Mach-Zehnder interferometer and hence the clear fringes obtained at 45◦ are washed out completely at
90◦.
VI. Discussion and Summary
In this experiment, interference characteristics at the single photon level were studied. An attenuated CW
laser was used to carry out Young’s double slit experiment with one photon at a time. Interference fringes
were observed which washed out as the acquisition time of the image was decreased. Single events confirming
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45º
50º
55º
60º
65º
70º
75º
80º
85º
90º
Figure 6: Fringe Visibility for different Polarizer D angles (45◦ to 90◦)
particle-like behavior of the photons were also observed. The Mach-Zehnder interferometer was used to carry
out interference between single photons using two paths instead of two slits. A polarizer was used after the
Mach-Zehnder interferometer to project the states onto the polarizer basis. This erased or preserved the
”which-path” information and hence the visibility was seen to have a strong dependence on the measurement
basis. Hence, interference was confirmed at the single photon level.
References:
(1) B. Greene, ”The Elegant Universe,” Vintage (2000) p 104.
(2) P. A. M. Dirac, ”The Principles of Quantum Mechanics,” 4th ed., OUP (1958) p 9.
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